HU-EP-14/65 HU-MATH-14/39 ZMP-HH-14/26

Prepared for submission to JHEP

**Double Wick rotating Green-Schwarz strings**

**Gleb Arutyunov**^{a,1}**and Stijn J. van Tongeren**^{b}

*a**Institut f¨**ur Theoretische Physik, Universit¨**at Hamburg, Luruper Chaussee 149, 22761 Hamburg,*
*Germany*

*a**Zentrum f¨**ur Mathematische Physik, Universit¨**at Hamburg, Bundesstrasse 55, 20146 Hamburg,*
*Germany*

*b**Institut f¨**ur Mathematik und Institut f¨**ur Physik, Humboldt-Universit¨**at zu Berlin, IRIS Geb¨**aude,*
*Zum Grossen Windkanal 6, 12489 Berlin, Germany*

*E-mail:* gleb.arutyunov@desy.de,svantongeren@physik.hu-berlin.de

Abstract:Via an appropriate field redefinition of the fermions, we find a set of conditions
under which light cone gauge fixed world sheet theories of strings on two different back-
grounds are related by a double Wick rotation. These conditions take the form of a set of
transformation laws for the background fields, complementing a set of transformation laws
for the metric and B field we found previously with a set for the dilaton and RR fields,
and are compatible with the supergravity equations of motion. Our results prove that at
least to second order in fermions, the AdS_{5} ×S^{5} mirror model which plays an important
role in the field of integrability in AdS/CFT, represents a string on ‘mirror AdS_{5}×S^{5}’,
the solution of supergravity that follows from our transformations. We discuss analogous
solutions for AdS_{3}×S^{3}×T^{4} and AdS_{2}×S^{2}×T^{6}. The main ingredient in our derivation is
the light cone gauge fixed action for a string on an (almost) completely generic background,
which we explicitly derive to second order in fermions.

1Correspondent fellow at Steklov Mathematical Institute, Moscow.

**Contents**

**1** **Introduction** **1**

**2** **Wick rotated bosons and changed geometry** **3**

**3** **Green-Schwarz fermions** **5**

**4** **Wick rotated fermions and changed background fields** **6**

4.1 Double Wick rotated Green-Schwarz fermions 7

4.2 The transformed Lagrangian 8

4.3 Fermions with metric and B field 10

4.4 Fermions with dilaton and RR fields 11

**5** **Double Wick related string backgrounds** **12**

**6** **Conclusions and outlook** **16**

**A Appendices** **18**

A.1 Γ matrices,*κ* symmetry, and other conventions 18

A.2 Light cone gauge fixing 20

A.3 The spin connection 23

A.4 Type IIB supergravity 24

**1** **Introduction**

A double Wick rotation on the world sheet of a string is a transformation that has proven
its use in the context of the AdS/CFT correspondence [1]. Fixing a light cone gauge in
the world sheet theory of a superstring on AdS_{5}×S^{5} [2] results in a model that transforms
nontrivially under a double Wick rotation, and it is the thermodynamics of this different
two dimensional quantum field theory [3, 4] that lies at the heart of the solution of the
spectral problem in AdS/CFT using integrability [5, 6]. Namely, the spectrum of the
string on AdS_{5}×S^{5} can be computed by means of the thermodynamic Bethe ansatz for
this so-called mirror model [7–10].^{1}

We may wonder whether such double Wick rotated models have a more direct physical interpretation. In previous work we showed that (given mild restrictions) doing a double Wick rotation at the bosonic level is equivalent to a particular transformation of the metric and B field [14]. Here we will establish the analogous result for the fermions, and find the

1These equations have been used to get impressive finite coupling results (see e.g. [11]), and have been simplified to the so-called quantum spectral curve [12], particularly impressive in the perturbative regime (see e.g. [13]).

transformation of (a combination of) the dilaton and Ramond-Ramond (RR) background fields such that the complete transformation is equivalent to a double Wick rotation of the Green-Schwarz action (to second order in fermions).

In order to find this transformation, we take a type IIB Green-Schwarz string to second
order in fermions on an (almost) completely generic background with light cone isometries,
and fix a light cone gauge. We then specify our restrictions on the metric and B field and
get the set of fermionic terms that are to be analytically continued. As fermions have single
derivative kinetic terms, their double Wick rotation is subtle and requires a complex field
redefinition; fortunately this can be unambiguously fixed in the flat space limit. We then
proceed by attempting to ‘undo’ the double Wick rotation, knowing the transformation
that accomplishes this at the bosonic level. Hence we take the general double Wick rotated
action and rewrite it in terms of the transformed metric and B field, changing the light
cone block of the metric as *g*lc → *g*lc*/|*det*g*lc| and the sign of the (transverse) B field.

As an immediate consistency check we find that all fermionic terms fixed by the metric
and B field directly match the original action as they must. We then compare the double
Wick rotated dilaton–RR terms (with transformed metric and B field) to the originals and
read off how they should transform. For our considerations to apply straightforwardly, we
find that similarly to how we assume our metric and B field not to mix light cone and
transverse directions, the RR fields should not lead to such mixing either. This translates
to the requirement that the RR fields should always have a single light cone index. The
dilaton and allowed RR fields then need to transform such that the combinations*e*^{Φ}*F*_{a(bcde)}^{(1/5)}
are invariant, while*e*^{Φ}*F*_{abc}^{(3)}picks up a sign, like*dB. Of course these RR fields now also live*
in a different space, so that the unchanging nature of *e*^{Φ}*F* is only apparent. While we do
not work out the details, the same analysis can be readily repeated for the type IIA string.

Then, by tracing the world sheet double Wick rotation back through the light cone
gauge fixing procedure,^{2} we discuss how to interpret the transformation of the background
fields from a target space perspective, as a combination of T dualities and target space
analytic continuation (Wick rotation). Though the analytic continuation is somewhat
subtle, this shows that our transformation is compatible with the supergravity equations
of motion, precisely under our restrictions on the background fields. Our main example is
AdS_{5}×S^{5} and its cousin ‘mirror AdS_{5}×S^{5}’, whose supergravity solution we constructed
in [14]. The associated background fields match our transformations, thereby explicitly
proving (to second order in the fermions) that the AdS_{5} ×S^{5} mirror model physically
represents a string on mirror AdS_{5}×S^{5}. We similarly briefly discuss mirror solutions for
AdS_{3} ×S^{3}×T^{4} and AdS_{2} ×S^{2} ×T^{6}. While our restrictions on the background do not
appear too drastic, spaces such as e.g. AdS_{4}×CP^{3} do not fit them. In the outlook we
discuss the apparent obstacles that arise when we relax our restrictions.

We should note that our work was ultimately inspired by the ‘mirror duality’ [15]

of the integrable deformation of the AdS_{5} ×S^{5} coset model of [16, 17].^{3} Labeling the
deformation by a parameter*κ, an exact S-matrix approach to this model showed that the*

2We would like to thank A. Tseytlin for this suggestion

3For further developments in this area see e.g. [18–30].

light cone theory at 1/κis equivalent to the double Wick rotation of the theory at *κ* [15].

This statement has a simple proof at the bosonic level using the metric and B field found
in [18], and in the *κ* → 0 limit gives precisely (mirror) AdS_{5} ×S^{5} [14]. The results of
our paper should allow us to check this statement at the fermionic level, assuming this
model corresponds to a string sigma model. However, to date the dilaton and RR fields
of this model have not been found, and whether this and related models represent strings
for generic values of*κ* is an interesting open question. We will briefly come back to this in
our conclusions.

In the next section we will begin our discussion by summarizing the transformation of
the metric and B field that is equivalent to a double Wick rotation at the bosonic level. In
section3 we introduce fermions on the world sheet, and present the light cone gauge fixed
action derived in appendixA.2. We then discuss the double Wick rotation of fermions, the
matching of the metric and B field terms, and the transformations of and constraints on the
dilaton and RR fields in section4. Section5contains the discussion of our transformation
from a target space perspective, along with the examples of AdS_{5} ×S^{5}, AdS_{3} ×S^{3} ×T^{4},
and AdS_{2}×S^{2}×T^{6} and their mirror partners. We finish in section 6 by discussing open
questions and generalizations of our work.

**2** **Wick rotated bosons and changed geometry**

To set the stage, let us recall the change of geometry that is equivalent to doing a double Wick rotation on the world sheet of a bosonic light cone string [14].

The action describing bosonic string propagation on a generic background is given by
*S*≡ −^{T}_{2}

Z

dτdσL=−^{T}_{2}
Z

dτdσ (g_{mn}*dx*^{m}*dx** ^{n}* −

*B*

_{mn}*dx*

*∧*

^{m}*dx*

*)*

^{n}*,*

where*T* is the (effective) string tension. We will consider*d-dimensional backgrounds with*
coordinates{t, φ, x* ^{µ}*}, packaging

*t*and

*φ*in the (generalized) light cone combinations

*x*^{+}= (1−*a)t*+*aφ,* and *x*^{−}=*φ*−*t ,* (2.1)
where *a* is a free parameter.^{4} Further conventions are summarized in appendix A.1. We
take the metric to be of the block diagonal form

*g** _{mn}* =

*g*_{++} *g*+− 0
*g*+− *g*−− 0
0 0 *g**µν*

≡ *g*_{lc} 0
0 *g**µν*

!

*,* (2.2)

with components depending only on the transverse coordinates *x** ^{µ}*, and consider purely
transverse B fields. We will address the question of relaxing these restrictions in section6.

As discussed in more detail in appendix A.2, to fix a uniform light cone gauge we
impose^{5}

*x*^{+} =*τ ,* *p*−= 1*.* (2.3)

4At the level of gauge fixing, this parameter interpolates between the temporal gauge at*a*= 0 and the
conventional light cone gauge at*a*= 1/2 [31]. In [14] we took*a*= 0 for conciseness.

5We focus on the zero winding sector in cases where*φ*parametrizes a circle the string can wind around.

## L

^{(0)}

## L ˜

^{(0)}

*τ*→*i˜**σ,* *σ*→ −i˜*τ*

*g*lc→ _{|det}^{g}^{lc}_{g}

lc|*,* *B*→ −B

**Figure 1. A double Wick rotation as a change of geometry. At the bosonic level a double Wick**
rotation of the light cone world sheet theory of a string is equivalent to a change of the metric and
B field.

The resulting action takes the form
*S*^{(0)} =−T

Z

dτdσ ^{p}*Y*^{(0)}+*g*^{+−}*/g*^{++}−*x*˙^{µ}*x*^{0ν}*B*_{µν}^{}*,* (2.4)
where

*Y*^{(0)}= ( ˙*x*^{µ}*x*^{0}* _{µ}*)

^{2}−

*AC*

*g*

^{++}

*g*−−

(2.5) with

*A*= 1 +*g*−−*x*^{0µ}*x*^{0}_{µ}*,*

*C* = 1 +*g*^{++}*x*˙^{ν}*x*˙*ν**,* (2.6)

and dots and primes refer to derivatives with respect to time (τ) respectively space (σ).

Note that we work in units that put spatial and temporal derivatives on the same footing.

Now we readily see that a double Wick rotation of the world sheet coordinates

*τ* →*i˜σ,* *σ* → −i˜*τ ,* (2.7)
gives an action of the same form, with *g*−− exchanged for −g^{++} while leaving *g*^{+−}*/g*^{++}

unchanged,^{6} and with a change of sign on *B. This change of the metric is nothing but*
*g*_{lc}→*g*^{−1}_{lc} = *g*_{lc}

|det*g*lc|*,* (2.8)

which applies in any coordinate system in the light cone subspace. In particular it is simply
*g**tt* ↔ *g** ^{φφ}* when the original metric is diagonal in

*t*and

*φ.*

^{7}Put differently, the action is formally left invariant under a double Wick rotation combined with the transformations (2.8) and

*B*→ −B on the background. We have illustrated this idea in figure1. Keep in mind that this combined transformation interchanges

*A*and

*C*and leaves

*Y*

^{(0)}invariant.

At this stage it is not clear that the mirror space associated to a string background represents a string background itself. We will come back to this point in detail later, for now let us simply assume there are examples where this is the case. We would then like to see that the fermions of such strings behave appropriately under this transformation as well. Let us therefore add fermions to our string, and fix a light cone gauge again, so that we can determine the required transformation properties of the dilaton and the RR fields.

6Since*g*^{+−}*/g*^{++}=−g+−*/g*−−, this is equivalent to exchanging*g*^{+−}for*g*+−.

7We define*ds*^{2}=−g*tt**dt*^{2} (+2g*tφ**dtdφ) +**g**φφ**dφ*^{2}+*g**µν**dx*^{µ}*dx** ^{ν}*.

**3** **Green-Schwarz fermions**

As we are interested in string backgrounds with RR fields, we need to use the Green-
Schwarz formalism. Though a fully explicit construction is not known, the Lagrangian
describing propagation of a Green-Schwarz superstring in an arbitrary background can be
constructed second order by second order in the fermions, explicitly done up to fourth
order in [32].^{8} We will limit ourselves to second order, which will provide a nontrivial test
of our ideas. As AdS_{5}×S^{5} is one of our spaces of interest, we will focus on the type IIB
string. The Lagrangian (density) at quadratic order in the fermions is given by [32]

L^{(2)}* _{f}* =

*i*

^{}

*?e*

^{a}*θΓ*¯

*a*Dθ−

*e*

^{a}*θΓ*¯

*a*

*σ*

_{3}Dθ

^{}

*,*(3.1) where

*θ*= (θ

_{1}

*, θ*

_{2}) is a doublet of sixteen component Majorana-Weyl spinors, with

*σ*

_{3}and the≡

*iσ*2 term inD (see below) acting in this two dimensional space. Dis given by

D=*d*−^{1}_{4}*ω/* +^{1}_{8}*e*^{a}*H** _{abc}*Γ

^{bc}*σ*

_{3}+

^{1}

_{8}

*e*

*SΓ*

^{b}

_{b}*,*(3.2) where

*ω*is the spin connection,

*e*the (one form) vielbein, and

*H*=

*dB*is the Neveu- Schwarz–Neveu-Schwarz (NSNS) three form. Slashes denote contraction with the appro- priate product of Γ matrices, Γ

*. S contains the dilaton and RR fields*

_{a...m}S =−e^{Φ}( /*F*^{(1)}+_{3!}^{1}*σ*_{1}*F/*^{(3)}+_{2·5!}^{1} * /F*^{(5)}), (3.3)
with the*F*^{(n)} denoting the*n*form RR fields, and Φ the dilaton. In components this reads
L^{(2)}* _{f}* =

*iγ*

^{αβ}*∂*

*α*

*x*

^{m}*θΓ*¯

*D*

_{m}

_{β}*θ*−

*i*

^{αβ}*∂*

*α*

*x*

^{m}*θΓ*¯

_{m}*σ*

_{3}D

_{β}*θ ,*(3.4) with

D* _{β}* =

*∂*

*β*−

^{1}

_{4}

*ω/*

_{m}*∂*

*β*

*x*

*+*

^{m}^{1}

_{8}

*∂*

*β*

*x*

^{m}*H*

*Γ*

_{mnp}

^{np}*σ*3+

^{1}

_{8}SΓ

_{m}*∂*

*β*

*x*

^{m}*.*(3.5) We prefer to write out the Lagrangian as

L=*γ** ^{αβ}*ˆ

*g*

_{mn}*∂*

_{α}*x*

^{m}*∂*

_{β}*x*

*−*

^{n}

^{αβ}*B*ˆ

_{mn}*∂*

_{α}*x*

^{m}*∂*

_{β}*x*

^{n}+*iγ*^{αβ}*∂*_{α}*x*^{m}*θΓ*¯ _{m}*∂*_{β}*θ*−*i*^{αβ}*∂*_{α}*x*^{m}*θΓ*¯ _{m}*σ*_{3}*∂*_{β}*θ ,* (3.6)
where the hats indicate these terms are not purely the bosonic metric and B field, i.e.

ˆ*g** _{mn}* =

*g*

*−*

_{mn}_{4}

^{i}*θΓ*¯

_{(m}

*ω/*

_{n)}*θ*+

_{8}

^{i}*θΓ*¯

_{(m}

*H*

*Γ*

_{n)pq}

^{pq}*σ*3

*θ*+

_{8}

^{i}*θΓ*¯

_{(m}SΓ

_{n)}*θ,*

*B*ˆ

*=*

_{mn}*B*

*−*

_{mn}_{4}

^{i}*θΓ*¯

_{[m}

*ω/*

_{n}_{]}

*σ*3

*θ*+

_{8}

^{i}*θΓ*¯

_{[m}

*H*

_{n}_{]pq}Γ

^{pq}*θ*+

_{8}

^{i}*θσ*¯ 3Γ

_{[m}SΓ

_{n]}*θ,*

(3.7)
with round and rectangular brackets denoting symmetrization and antisymmetrization re-
spectively, defined with the usual factor of 1/n!. In this notation we will treat ˆ*g* as if it
really were the metric, so it can be used to raise and lower indices at intermediate stages
of computation.^{9} When light cone gauge fixing, the fermions that contribute to terms in

8To second order this Lagrangian was first derived in [33,34].

9The price of this hopefully unambiguous notation is having to keep track of extra signs in fermionic terms in the inverse ‘metric’. Note that while not explicitly done in this paper, this notation is very useful for example when fixing a light cone gauge in a Hamiltonian setting.

*g*ˆand ˆ*B* that are already nonzero at the bosonic level formally go along for the (bosonic)
ride, while possible extra nonzero terms need only be kept at a linearized level (quadratic
in fermions). Of course we still need to add the manifestly fermionic terms in eq. (3.6) to
the derivation. The only assumption we will make at this stage is that the fermions do not
generate a nonzero ˆ*B*+−, which will turn out to follow from our restrictions on the metric
and B field and the restrictions on the RR fields we will find later.

**Light cone action**

In appendixA.2we fix a light cone gauge for a string on a completely generic background
with light cone isometries and*B*+− = 0, including fermions to second order. For the type
of backgrounds we are considering, this gives the gauge fixed action

*S*=*S*^{(0,2)}−*T*
Z

dτdσ^{}L^{(2)}* _{a}* +L

^{(2)}

_{b}^{}

*,*(3.8) where

*S*

^{(0,2)}is the bosonic gauge fixed action (2.4) with metric and B field replaced by (the relevant parts of) ˆ

*g*and ˆ

*B.*L

^{(2)}

*a*and L

^{(2)}

*are given by*

_{b}L^{(2)}* _{a}* = 1
2

√
*Y*^{(0)}

1
*g*−−*g*^{++}

h

(2ˆ*g*^{+µ}*x*˙*µ*−*iθΓ*¯ ^{+}*θ)A*˙ + (2 ˆ*B*−µ*x*^{0µ}+*iθΓ*¯ −*σ*3*θ*^{0})C (3.9)

−^{}*g*−−(2ˆ*g*^{+µ}*x*^{0}* _{µ}*−

*iθΓ*¯

^{+}

*θ*

^{0}) +

*g*

^{++}(2 ˆ

*B*−µ

*x*˙

*+*

^{µ}*iθΓ*¯ −

*σ*

_{3}

*θ)*˙

^{}

*x*˙

_{µ}*x*

^{0µ}

^{i}and

L^{(2)}* _{b}* = − 1

*g*−−

*i*
2

*θΓ*¯ −*θ*˙+ ˆ*g*−µ*x*˙^{µ}

− 1
*g*^{++}

*i*
2

*θΓ*¯ ^{+}*σ*_{3}*θ*^{0}+ ˆ*B*^{+µ}*x*^{0}_{µ}

(3.10)
which are the formally new terms introduced by the fermions. *A* and *C* are those of eqs.

(2.6). In this action we have fixed a *κ* symmetry gauge by taking

(Γ^{0}+ Γ* ^{p}*)θ= 0, (3.11)

where Γ^{0} and Γ* ^{p}* are the flat space cousins of Γ

*and Γ*

^{t}*, see appendixA.1. This means we essentially dropped terms of the form ¯*

^{φ}*θΓ*

^{µ}*θ*in Γ matrix structure; appendix A.2 contains the full action before imposing a

*κ*symmetry gauge choice.

**4** **Wick rotated fermions and changed background fields**

In section2we recalled how at the bosonic level a double Wick rotation on the world sheet is actually equivalent to a change of background. We would now like to investigate this at the fermionic level. To do so, we will proceed as before, carefully considering a double Wick rotation of the general action (3.8), and attempting to reinterpret the result as an action of the same form.

The story is slightly more involved for fermions than for bosons, due to their single derivative kinetic terms. Already for the massless free Dirac Lagrangian it is clear that a double Wick rotation does not result in a real Lagrangian if we keep the conjugation properties of the fermions fixed. Hence we need to accompany a double Wick rotation by a

## L L ˜ L ˇ

*τ* →*i˜**σ,* *σ*→ −i˜*τ*

*g*lc→ _{|}_{det}^{g}^{lc}_{g}

lc|

*B*→ −B
S →S˜

**Figure 2. A double Wick rotation as a change of background fields? Combining a double Wick**
rotation with a change of the metric and B field and insisting the result agrees with the original
action should fix the transformationS →S˜of the dilaton and RR fields. The intermediate ˇLhas
no obvious physical interpretation.

change of reality condition, or equivalently, a complex field redefinition.^{10} We will fix this
field redefinition by considering a Green-Schwarz string in flat space, which we will then
use to find the double Wick rotated action on a general background.

To reinterpret the result as an action of the form (3.8) again, we will attempt to com- plete the bosonic diagram of figure1to the one of figure2, finding the total transformation of the background fields that ‘undoes’ the double Wick rotation.

**4.1** **Double Wick rotated Green-Schwarz fermions**

To determine the precise transformation of our fermions under a double Wick rotation, we
will consider a Green-Schwarz string in flat space. The light cone Lagrangian for such a
string is given by^{11}

L= *T*
2

*x*˙^{µ}*x*˙*µ*−*x*^{0µ}*x*^{0}* _{µ}*+

*iθΓ*¯

^{−}

*θ*˙+

*iθΓ*¯

^{−}

*σ*3

*θ*

^{0}

^{}

*.*(4.1) Now let us carefully do a double Wick rotation

*x(τ, σ)*→*x(i˜σ,*−i˜*τ*)≡*y(˜τ ,σ)*˜

*θ(τ, σ)*→*θ(i˜σ,*−i˜*τ*)≡*θ(˜*˜*τ ,σ)*˜ (4.2)
so that we have

*x*^{0} =*iy,*˙ *x*˙ =−iy^{0}*,*
*θ*^{0} =*iθ,*˙˜ *θ*˙=−i*θ*˜^{0}*,*

(4.3) where dots and primes refer to derivatives in the first and second arguments. The resulting action then takes the form

L= *T*
2

*y*˙^{µ}*y*˙* _{µ}*−

*y*

^{0µ}

*y*

^{0}

*+*

_{µ}*θΓ*¯˜

^{−}

*θ*˜

^{0}−

*θΓ*¯˜

^{−}

*σ*

_{3}

*θ*˙˜

^{}

*.*(4.4)

10In the context of the AdS5×S^{5} coset sigma model this was noted explicitly in [4], here we will be
considering a transformation of the canonical Green-Schwarz fermions in a general setting.

11This can be obtained from our general action in flat space, taking the limit*a*→1/2 from below. Note
that in flat space with*a*= 1/2, Γ^{+}=*G*^{+}, where*G*^{+} is the matrix involved in*κ*symmetry gauge fixing as
discussed in appendixA.1.

We see that the bosonic part of this Lagrangian is formally the same as that of the original
Lagrangian (4.1), and we can assume *y* to have the reality properties of *x; typically they*
are not even distinguished. If we however assume ˜*θ(˜τ ,*˜*σ) to have the reality properties of*
*θ(τ, σ), this action will not be real when the original Lagrangian is. To fix this we should*
change the reality properties of ˜*θ. Equivalently, we would like to consider a (constant,*
complex) field redefinition

*θ*˜=*M η,* (4.5)

where the action is real when written in terms of*η* with conventional reality properties. A
priori it is not clear how to fix*M* exactly, but we have an additional physical requirement
to impose.

As a light cone string in flat space is a Lorentz invariant model (manifest in the NSR description), a double Wick rotation should leave all physical properties of the model invariant. The simplest way to realize this is to insist that the form of the Lagrangian is invariant under a double Wick rotation. Comparing eqs. (4.1) and (4.4), we see that this is the case if

*ηM*¯ * ^{t}*Γ

^{−}

*M η*

^{0}=

*i¯ηΓ*

^{−}

*σ*

_{3}

*η*

^{0}

*,*(4.6) and

*ηM*¯ * ^{t}*Γ

^{−}

*σ*3

*Mη*˙=−i¯

*ηΓ*

^{−}

*η.*˙ (4.7) Not having or wanting to touch the Γ matrices we need

*M*^{t}*M* =*iσ*3 *M*^{t}*σ*3*M* =−i1. (4.8)

This fixes

*M* = 0 *b*
*c* 0

!

*,* (4.9)

with*c*^{2} =−b^{2}=*i, mixing the two Majorana-Weyl spinors. The overal sign ofM* is clearly
inconsequential, but this still leaves the choice of *bc* = ±1. Later we will see that this
choice is actually inconsequential as well. In summary, a double Wick rotation can be
implemented by the replacements

*x*^{0} →*ix,*˙ *x*˙ → −ix^{0}*,*

*θ*→*M η,* *θ*^{0} →*iMη,*˙ *θ*˙→ −iM η^{0}*,* (4.10)
where we have stopped carefully distinguishing the bosons.

**4.2** **The transformed Lagrangian**

Looking at figure2it is clear that we are primarily interested in ˇL, the result of transforming
L by combining a double Wick rotation with a change of the metric and B field so that
the bosonic part of the action is left invariant. This change comes with strings attached
however, since it appears hard to give a target (or flat) space interpretation to the result
of transforming the metric in a target space Γ matrix; for a diagonal metric, Γ* ^{t}*=

^{p}

*g*

*Γ*

^{tt}^{0}

becomes √

*g** _{φφ}*Γ

^{0}for example. This can be fixed by the automorphism of our Clifford algebra taking

Γ^{0/p}→*iΓ*^{p/0}*,* (4.11)

where *p* is the other flat light cone direction, associated to *φ.* This automorphism is
compatible with our*κ*symmetry gauge fixing, and is equivalent to a unitary change of basis
by *e*^{−i}^{π}^{4}^{Γ}^{0}^{Γ}* ^{p}* on the fermions.

^{12}Combining the change of metric with this automorphism we get

^{13}

Γ^{±}
Γ±

!

→ Γˇ^{±}
Γˇ±

!

→ ±i Γ_{∓}

−Γ^{∓}

!

*,* (4.12)

where the rightmost Γ matrices now have proper target space indices with respect to the
changed metric. The factors of *i* will disappear again since thanks to our *κ* symmetry
gauge we also have

*η*¯→*i¯η.* (4.13)

Taking the action (3.8) and doing the above double Wick rotation, changing the metric and B field and implementing this automorphism we get

*S*= ˇ*S*^{(0,2)}−*T*
Z

d˜*τ*d˜*σ*^{}Lˇ^{(2)}* _{a}* + ˇL

^{(2)}

_{b}^{}

*,*(4.14) with

Lˇ^{(2)}* _{a}* = 1
2

√
*Y*^{(0)}

1
*g*−−*g*^{++}

h(i2*B*ˇˆ−µ*x*˙* ^{µ}*−

*i¯ηΓ*

^{+}

*η)A*˙ + (−i2ˇˆ

*g*

^{+µ}

*x*

^{0}

*+*

_{µ}*i¯ηΓ*−

*σ*

_{3}

*η*

^{0})C (4.15)

−

*g*−−(i2*B*ˇˆ−µ*x*^{0µ}−*i¯ηΓ*^{+}*η*^{0}) +*g*^{++}(−i2ˇˆ*g*^{+µ}*x*˙* _{µ}*+

*i¯ηΓ*−

*σ*

_{3}

*η)*˙

*x*˙_{µ}*x*^{0µ}^{i}
and

Lˇ^{(2)}* _{b}* = − 1

*g*−−

*i*

2*ηΓ*¯ −*η*˙−*iB*ˇˆ^{+µ}*x*˙*µ*

− 1
*g*^{++}

*i*

2*ηΓ*¯ ^{+}*σ*_{3}*η*^{0}+*ig*ˇˆ−µ*x*^{0µ}

*.* (4.16)

Remaining checks denote transformed quantities we have temporarily left implicit. Com-
paring this to the general action (3.8), we see that the manifestly fermionic terms match
perfectly with*η*=*θ. From the remaining terms we read off that we need to have*

ˇˆ

*g*^{+µ}=*iB*ˆ_{−}^{µ}*,* *B*ˇˆ_{−}* ^{µ}*=−iˆ

*g*

^{+µ}

*,*ˇˆ

*g*−µ=−i*B*ˆ^{+}_{µ}*,* *B*ˇˆ^{+}* _{µ}*=

*iˆg*−µ

*.*

(4.17)
We also have to check that the fermionic terms in ˇ*S*^{(0,2)} behave as the metric and B field,
i.e. *g*ˇˆ−−=−ˆ*g*^{++}*,* *g*ˇˆ^{++}=−ˆ*g*−−*,*

ˇˆ

*g*+−= ˆ*g*^{+−}*,* *g*ˇˆ^{+−}= ˆ*g*+−*,*
ˇˆ

*g** _{µν}* = ˆ

*g*

_{µν}*,*

*B*ˇˆ

*=−*

_{µν}*B*ˆ

_{µν}*.*

(4.18)

12Of course, in our*κ*symmetry gauge this is just multiplication by*e*^{−i}^{π}^{4}, and actually does nothing to
our expressions. This is not a particularly useful point of view however.

13Restricting ourselves to the 2d light cone block, starting with the original vielbein*e*satisfying*e*^{t}*ηe*=*g*
we can construct a ‘mirror’ vielbein as ˜*e*=*g*^{−1}*eσ*1, since the mirror metric is just*g*^{−1}*. This means we*
have*g*^{−1}*eηΓ =**e σ*˜ 1*ηΓ, i.e.* *e*^{±}* _{a}*Γ

*=∓˜*

^{a}*e*

*∓(σ1)*

^{a}

_{a}

^{b}*η*

*bc*Γ

*. Combined with the transformation Γ→*

^{c}*iσ*1Γ we get the desired result.

Finally, in order not to leave our present framework, the assumption ˆ*B*+−= 0 needs to be
preserved. We will begin by checking that the fermionic terms in ˆ*g* and ˆ*B* determined by
the metric and B field actually behave as they should.

**4.3** **Fermions with metric and B field**

The fermionic terms in ˆ*g* and ˆ*B* that have only the metric or B field in them are ¯*θΓ*_{m}*ω/*

*n**θ*
and ¯*θΓ*_{m}*H** _{npq}*Γ

^{pq}*σ*3

*θ, with an extra*

*σ*3 inserted for ˆ

*B. Our*

*κ*symmetry gauge choice together with our restrictions on the metric and B field imply that many of these terms vanish. Using the block notation of eq. (2.2) we have

*θΓ*¯ _{m}*ω/*_{n}*θ*=
0 •

• 0

*mn*

*,* (4.19)

where bullet points denote asymmetric (generically) nonzero contributions. In other words
the spin connection term in ˆ*g*and ˆ*B*only contributes to ˆ*g*±µand ˆ*B*±µ, which by assumption
have no bosonic term. *H* contributes similarly

*θΓ*¯ _{m}*H** _{npq}*Γ

^{pq}*θ*= 0 •

• 0

*mn*

*.* (4.20)

These terms therefore only contribute to the relations (4.17).

The first of eqs. (4.17) requires the spin connection terms to transform as

*θΓ*¯ ^{(+}*ω/*^{µ)}*θ*→ −i*θΓ*¯ _{[−}*ω/*^{µ]}*σ*_{3}*θ* (4.21)
under the full set of transformations used to arrive at ˇL, since the fermionic contribution
to the inverse of the ‘metric’ ˆ*g* has raised indices but an opposite sign. Now under these
transformations the light cone components of the spin connection change as

*/*
*ω*^{±}
*ω/*_{±}

!

→ ∓i *ω/*_{∓}

−*ω/*^{∓}

!

*,* (4.22)

while *ω/** _{µ}* is unaffected, see appendixA.3 for details. Combining this with all other trans-
formations we get

*θ*¯^{}Γ^{+}*ω/** ^{µ}*+ Γ

^{µ}*ω/*

^{+}

^{}

*θ*→(i¯

*η)*

^{}(iΓ−)

*ω/*

*+ Γ*

^{µ}*(−i/*

^{µ}*ω*

_{−})

^{}(iσ

_{3})η=−i¯

*η*

^{}Γ

_{−}

*ω/*

*−Γ*

^{µ}

^{µ}*ω/*

_{−}

^{}

*σ*

_{3}

*η,*nicely matching eq. (4.21) upon identifying

*η*=

*θ. The rest of eqs. (4.17) works analo-*gously, the relative sign arising cf. eqs. (4.8).

The*H*contribution to these relations has an extra*σ*_{3} but otherwise proceeds similarly.

The last of eqs. (4.17) for example requires

*θΓ*¯ ^{[+}*H** _{µ]νρ}*Γ

^{νρ}*θ*→

*iθΓ*¯

_{(−}

*H*

*Γ*

_{µ)νρ}

^{νρ}*σ*

_{3}

*θ,*while our transformation gives

*θΓ*¯ ^{+}*H** _{µνρ}*Γ

^{νρ}*θ*→(i¯

*η)(iΓ*−)(−H

*)Γ*

_{µνρ}*(iσ*

^{νρ}_{3})η=

*i¯ηΓ*−

*H*

*Γ*

_{µνρ}

^{νρ}*σ*

_{3}

*η*matching precisely since

*H*is purely transverse.

**4.4** **Fermions with dilaton and RR fields**

Having checked that the metric and B field terms behave consistently, let us investigate
the fermionic terms involving the dilaton and RR fields. A priori these terms can give
contributions to any part of ˆ*g* and ˆ*B. If our transformation is to work we get immediate*
restrictions however, due to the transformation properties of the spinor structure inS (see
eq. (3.3))

*M*^{t}*M*=−bc, *M*^{t}*σ*1*M* =*bcσ*1*,* (4.23)

actually leaving the type of couplings invariant. Moreover these transformations guarantee
that a real*iθΓ*¯ * _{M}*SΓ

_{n}*θ*remains real, since

*bc*=±1 and fermionic terms that are nonzero in our

*κ*gauge contain an even number of non-transverse Γ matrices. This makes it impossible for the contribution ofS to relate the double Wick rotation of ˆ

*g, to ˆB, and vice versa.*

^{14}

From the fact that our forms cannot lead to a mixing of light cone and transverse
components in ˆ*g* and ˆ*B,*

*θΓ*¯ ^{±}SΓ^{µ}*θ*= ¯*θΓ** ^{µ}*SΓ

^{±}

*θ*= 0

*,*(4.24) we deduce that each nonzero term in S must have a single light cone index, since three or more light cone indices necessarily give zero, and two or none gives a nonzero contribution in the above. Note that these conditions also immediately imply ˆ

*B*+− = 0. We then have to insist these terms transform to match eqs. (4.18). For the light-cone components we

have *θΓ*¯ ^{+}SΓ^{±}*θ*

*θΓ*¯ ^{−}SΓ^{±}*θ*

!

→ ±i −*θΓ*¯ −SΓˇ _{∓}*θ*
*θΓ*¯ _{+}SΓˇ ∓*θ*

!

*,* (4.25)

where

Sˇ=−bcS^{}^{}

_{Γ}^{0/5}_{→iΓ}^{5/0}

*σ*1→−σ1

*.* (4.26)

The purely transverse components simply give

*θΓ*¯ * _{µ}*SΓ

_{ν}*θ*→

*iθΓ*¯

*SˇΓ*

_{µ}

_{ν}*θ.*(4.27) In the ˆ

*B*case we get an extra sign here due to

*σ*3. Taking into account the extra minus sign for the fermions in the inverse of ˆ

*g, these transformations are compatible with eqs.*

(4.18) provided we have a set of mirror fields (packaged in ˜S) such that
*θΓ*¯ * _{m}*SΓ˜

_{n}*θ*=

*iθΓ*¯ * _{m}*SΓˇ

_{n}*θ*

*m,n*∈ {µ},

−i*θΓ*¯ * _{m}*SΓˇ

_{n}*θ*

*m,n*∈ {+,−}. (4.28) The problematic looking relative sign we can now remove by inserting 1 in the form of (Γ

^{0}Γ

*)*

^{p}^{2}, giving

*θΓ*¯ * _{m}*SΓ˜

_{n}*θ*=−i

*θΓ*¯

*Γ*

_{m}^{0}Γ

*SΓˇ*

^{p}

_{n}*θ*(4.29)

14Since *σ*1*σ*3 =−and in our *κ*symmetry gauge we have Γ^{0}Γ^{p}*θ* =*θ, we might imagine that (the 0p*
containing components of) an *F*^{(n)} term effectively starts looking like an *F*^{(n−2)} term with an extra*σ*3.
However these terms would not couple correctly to the bosons (sitting in ˆ*g*rather than ˆ*B* or vice versa),
and moreover would still not pick up the required factor of *i. Still, we could more specifically imagine a*
pair of forms such that*F**/*^{(n)}=±*F**/*^{(n−2)}Γ^{0}Γ* ^{p}*, giving

*σ*1±

*Γ*

^{0}Γ

*which removes terms due to our*

^{p}*κ*gauge choice. In this particular setting the contributions to ˆ

*g*and ˆ

*B*take the same overall form. Nevertheless the lacking factor of

*i*remains a problem.

since Γ^{0}Γ* ^{p}* commutes with transverse Γ matrices but anti-commutes with light cone ones,
and we have ¯

*θΓ*

^{0}Γ

*= −*

^{p}*θ. This means that everything is precisely compatible, provided*¯ the forms of the mirror background are given by

S˜=−iΓ^{0}Γ* ^{p}*Sˇ=−ibcΓ

^{0}Γ

*S*

^{p}^{}

^{}

_{}

Γ^{0/5}→iΓ^{5/0}
*σ*1→−σ_{1}

*.* (4.30)

We can actually rewrite this very nicely. Since S must have one and only one light cone
index, we can always write it as Γ^{0}*N* + Γ^{p}*K* for some*N* and *K* that do not contain Γ^{0} or
Γ* ^{p}*. In this form it is clear however that multiplication by Γ

^{0}Γ

*undoes the interchange of Γ*

^{p}^{0}for Γ

*in ˇS, leaving just an extra factor of*

^{p}*i. In short, we have*

S˜=−iΓ^{0}Γ* ^{p}*Sˇ=

*bcS*

^{}

^{}

_{}

*σ*1→−σ_{1}*.* (4.31)

This transformation is uniquely fixed up to the sign choice *bc* = ±1. However, since a
simultaneous sign change on all RR fields leaves the supergravity equations of motion [35]

invariant, we can consider this sign choice irrelevant anyway.

We see that two backgrounds are related by a double Wick rotation provided that the metric and B field are related as in figure 1 and the dilaton and RR fields as

*e*^{Φ}*F/*^{(n)}→(i)^{n−1}*e*^{Φ}*F/*^{(n)}*,* (4.32)
where we have chosen *bc* = 1. With this choice, the story unifies at the level of the
underlying even degree forms (see appendix A.4)

*e*^{Φ}*d /C*^{(n)}→*i*^{n}*e*^{Φ}*d /C*^{(n)}*,*
*d /B*→*i*^{2}*d /B.*

(4.33)
In short, up to a possible sign the combinations*e*^{Φ}*F/* for the mirror background are identical
to the originals. Of course the slashes can be removed, but then it is important to note
that the relations hold between tensors with flat space indices. Our full story can now be
summarized by figure 3. At this level we cannot immediately disentangle the dilaton and
the RR fields.

**5** **Double Wick related string backgrounds**

In the previous section we found transformation laws for the background fields via an explicit double Wick rotation on the world sheet combined with an appropriate fermionic field redefinition. Before giving a few examples of pairs of backgrounds related by these transformations, let us discuss how these transformation laws can also be viewed more directly from a target space point of view.

As used in our appendix, light cone gauge fixing can be viewed as T dualizing in the*x*^{−}
direction, and gauge fixing the corresponding T dual field*ψ*=*σ*in addition to*x*^{+}=*τ* [36].

It should then be possible to (formally) view a double Wick rotation on the world sheet at
the target space level as a T duality in*x*^{−}, followed by the analytic continuation (x^{+}*, ψ)*→

## L L ˜

*τ*→*i˜**σ,* *σ*→ −i˜*τ*

*g*lc→ _{|det}^{g}^{lc}_{g}

lc|*,* *B*→ −B,
*e*^{Φ}*F**/*^{(n)}→(i)^{n−1}*e*^{Φ}*F**/*^{(n)}

**Figure 3. A double Wick rotation as a change of background fields. A double Wick rotation of**
the light cone gauge fixed world sheet theory of a Green-Schwarz string is equivalent to a change of
background fields, at least to second order in the fermions. It is not clear whether these changed
fields always correspond to a solution of supergravity.

(i*ψ,*˜ −i˜*x*^{+}), and finally another T duality in the ˜*ψ* direction.^{15} As this procedure involves
analytic continuation of target space fields as well as T duality in a light cone direction
however,^{16} this procedure takes us out of the realm of real supergravity and string theory,
and generically does not result in a real solution of supergravity. Furthermore, the details
of this analytic continuation are subtle, as the complex ‘diffeomorphism’ corresponding to
the double Wick rotation is ‘improper’, i.e.

*τ*˜(˜*x*^{+})
*σ( ˜*˜ *ψ)*

!

= 0 *i*

−i0

! *τ*(x^{+})
*σ(ψ)*

!

(5.1) is a transformation with determinant−1. From this point of view our transformation laws should be compatible with the supergravity equations of motion as follows.

Firstly we observe that precisely under our restrictions on the metric and B field,
and the ones just found for the RR fields, the combination of T dualities and analytic
continuation preserves reality of the background fields. Namely, T dualizing metrics and B
fields that fit our restrictions cannot lead to mixing of light cone and transverse directions,
which are the components that would pick up factors of *i*under the analytic continuation.

Similarly, given RR fields with a single light cone index, T duality in *x*^{−} results in (type
IIA) RR fields with either no *ψ* or + index or both, which hence remain real under the
analytic continuation.

This sequence of T duality, analytic continuation, and T duality, should match the
transformations we found based on our world sheet perspective. To avoid technical com-
plications, we will demonstrate this explicitly for metrics diagonal in *t* and *φ, and fixing*
the static gauge*t*=*τ*,*p** _{φ}*= 1.

^{17}By assumption our RR fields then only have components

15We would like to thank A. Tseytlin for this suggestion.

16In principle this light cone T duality can be avoided by considering the static gauge instead of our
generalized light cone gauge, leading to T dualities in the*φ*directions.

17This gauge is equivalent to our light cone gauge for*a*= 0.

involving*t*or*φ, and upon T dualizing* *φ*to*ψ* we get
*F*_{t...}

*F**φ...*

!

T duality

−→ *F*_{tψ...}

−F_{...}

!

*,* (5.2)

where the dots denote an even number of transverse indices. The analytic continuation
is slightly subtle however, involving more than the simple replacement (t, ψ) →(i*ψ,*˜ −i˜*t).*

The reason for this can be seen from the bosonic type IIA supergravity action (see e.g.

page 172 of [37]). While this action is generically diffeomorphism invariant, this involves
a little more work for improper diffeomorphisms such as ours, due to the Chern-Simons
term *B*∧*F*^{(4)}∧*F*^{(4)}. This term behaves like a pseudoscalar under diffeomorphisms, and
picks up a sign under our analytic continuation.^{18} We can fix this sign by combining our
analytic continuation with a sign flip on*B*. However, to then keep the kinetic term|*F*ˆ^{(4)}|^{2}
invariant, since ˆ*F*^{(4)} =*F*^{(4)}−*C*^{(1)}∧*H, we also need to flip the sign of eitherF*^{(4)} or*F*^{(2)}
(C^{(1)}). Importantly, this introduces a relative sign between *F*^{(n)} and *F*^{(n+2)}. The result
of the analytic continuation and a second T duality then becomes

*F*_{tψ...}

−F_{...}

!

an. cont.

−→ ± *F*_{˜}_{t}*ψ...*˜

−F_{...}

!

T duality

−→ ± *F*˜*t...*

*F**φ...*˜

!

*,* (5.3)

where in the analytic continuation step we implicitly swapped the indices for a sign, and the
overall±sign refers to the degree of the form, being plus for*F*^{(1)} and *F*^{(5)}, and minus for
*F*^{(3)}, or vice versa.^{19} Combining this with the sign flip of*B*under the analytic continuation,
and the dilaton generated by the double T duality that is clearly compatible with *e*^{Φ}*F/* =
*e*^{Φ}^{˜}*F, we see that this precisely matches the transformations we found previously from a*˜*/*
world sheet perspective, which are therefore compatible with the supergravity equations of
motion.

After this general discussion let us give some examples. While our restrictions on the
metric and B field do not appear too drastic, the number of explicitly known solutions of
supergravity is also not very large; we will consider AdS_{5}×S^{5}, AdS_{3}×S^{3}×T^{4} supported
by a RR three form, and AdS_{2}×S^{2}×T^{6}.^{20} We distinguish mirror space quantities by
tildes below.

**AdS**_{5}**×S**^{5}

The metric of AdS_{5}×S^{5} in global coordinates
*ds*^{2}= −(1 +*ρ*^{2})dt^{2}+ *dρ*^{2}

1 +*ρ*^{2} +*ρ*^{2}*dΩ*_{3}+ (1−*r*^{2})dφ^{2}+ *dr*^{2}

1−*r*^{2} +*r*^{2}*dΩ*_{3}*,* (5.4)

18To see this explicitly, note that our RR fields T dualized to type IIA have either both *t* and *ψ* as
components, or neither, where the contribution with both picks up a sign under analytic continuation.

19Recall that we encountered the same inconsequential sign freedom in our world sheet discussion above.

20To have a case with a transverse B fields we could for example consider the three parameter general-
ization of the Lunin-Maldacena background [38] constructed in [39], which fits our restrictions when only
*γ*1 is nonzero. Presumably the associated mirror space can be obtained by TsT transformations on mirror
AdS5×S^{5}; we will not pursue this interesting case here.

is precisely of the form discussed in section 2, and its mirror companion is [14]

*ds*˜^{2} = − 1

1−*r*^{2}*dt*^{2}+ *dρ*^{2}

1 +*ρ*^{2} +*ρ*^{2}*dΩ*_{3}+ 1

1 +*ρ*^{2}*dφ*^{2}+ *dr*^{2}

1−*r*^{2} +*r*^{2}*dΩ*_{3}*.* (5.5)
Note that this metric corresponds to a direct product of two manifolds with coordinates
*t,* *r, and associated angles, and* *φ,* *ρ, and associated angles. We have written the mirror*
metric with this nonstandard ordering of coordinates, so that we leave the labeling of the
transverse space untouched with respect to AdS_{5}×S^{5}.

Both these spaces can be embedded in type IIB supergravity, supported by a self dual
five form and dilaton. The relevant equations of motion are given in appendix A.4. For
AdS_{5}×S^{5} these equations are solved by a constant dilaton Φ = Φ_{0}, and a five form

1

4*e*^{Φ}*F/* = Γ^{01234}−Γ^{56789}*.* (5.6)

In line with the above discussion, mirror AdS_{5}×S^{5} [14] has a nontrivial dilaton
Φ = ˜˜ Φ_{0}−1

2log(1−*r*^{2})(1 +*ρ*^{2}), (5.7)
and its combination with the five form is precisely such that

1

4*e*^{Φ}^{˜}*F*˜*/* = Γ^{01234}−Γ^{56789}= ^{1}_{4}*e*^{Φ}*F ./* (5.8)
Note that since the metrics differ, the physical meaning of the forms is very different
between AdS_{5}×S^{5} and its mirror version; in the mirror case the form mixes directions
belonging to the two five dimensional submanifolds, while the form is just a difference of
five dimensional volume forms for AdS_{5}×S^{5}.

**AdS**_{3}**×S**^{3}**×T**^{4}**and AdS**_{2}**×S**^{2}**×T**^{6}

Taking our *φ* coordinate to be the equatorial angle on the sphere, the mirror metrics
associated to AdS_{3}×S^{3}×T^{4} and AdS_{2}×S^{2}×T^{6} are just the lower dimensional analogues
of eq. (5.5) completed to ten dimensions with flat directions. We choose to label coordinates
such that*φ*∼*n, hencep*=*n, for the AdS*_{n}×S^{n} case.

The supergravity equations of motion are solved by a constant dilaton for AdS_{3}×S^{3}×T^{4}
and AdS_{2}×S^{2}×T^{6}. We take AdS_{3}×S^{3}×T^{4} to be supported by the three form

1

2*e*^{Φ}*F/* = Γ^{012}+ Γ^{345}*,* (5.9)

while AdS_{2}×S^{2}×T^{6} is supported by the five form
*e*^{Φ}*F/* = Γ^{01}Re(*w/*

C^{3})−Γ^{23}Im(*w/*

C^{3}), (5.10)

where *w*_{C}^{3} is the holomorphic volume form on *T*^{6} in complex coordinates. Based on our
discussion above, this should then also give the solution for their mirror versions. Indeed,
we can readily check that this is the case with the mirror space dilaton in both cases given
by eq. (5.7).^{21}

21These mirror space solutions can be independently derived by making an obvious ansatz based on the
result for AdS5×S^{5}. Alternately, they can also be obtained by T dualizing in*t*and*φ, giving dS*2/3×H^{2/3}×
T^{6/4} which are simple solutions of type IIB^{∗}supergravity analogous to AdS2/3×S^{2/3}×T^{6/4} in type IIB
supergravity.